Application of Grey System Theory in Approximate Calculation of Wave Packet Evolution

TL;DR

This paper introduces a novel application of grey system theory, specifically the GM(1,1) model, to approximate the evolution of wave packets in quantum mechanics, offering a new way to handle uncertainty in physical parameters.

Contribution

It is the first to apply grey system theory to physical wave packet evolution, providing an approximate calculation method that accounts for uncertainties in parameters.

Findings

Successfully modeled wave packet evolution using GM(1,1)

Achieved error analysis and correction for the approximation

Demonstrated method with Gauss wave packet example

Abstract

The study of wave packet is of great significance in quantum mechanics, optics and fluid mechanics. However, in order to solve the strict evolution behavior of wave packet, it is necessary not only to determine the parameters of various physical quantities such as mass, but also to carry out the complex integration process in configuration space and momentum space. In the final analysis, the evolution behavior of wave packet is the evolution behavior of distribution function parameters with time. Using the method of grey system theory, the time-varying parameters are replaced by time-varying response series, and then the formal structure is constructed according to the physical meaning of time evolution to realize the approximate simulation and approximate calculation of wave function expression. The advantage of this method is that it can simulate the evolution of wave packet under the uncertainty of each physical quantity. Based on the grey system theory, this paper uses GM (1,1) model to replace the time response function of wave packet distribution, obtains the simulated evolution behavior of wave packet wave function by means of mathematical techniques such as variational method, and makes error analysis and correction. In practical application, the approximate evolution behavior of wave function can be obtained only by determining the parameters of expansion coefficient of wave function. In this paper, the Gauss wave packet is taken as an example to discuss the method. The grey system theory of mathematical modeling is introduced into physical calculation for the first time, and an approximate calculation method which can be widely used in wave packet evolution calculation is given.